Chapter 6. Quadrature - MathWorks

6.5. Performance 9
calls for linear algebraic vector operations that are not appropriate here. The Matlab
function vectorize transforms a scalar expression into something that can be
used as an argument to function functions.
Many of the function functions in Matlab require the specification of an
interval of the x-axis. Mathematically, we have two possible notations, a ≤ x ≤ b
or [a, b]. With Matlab, we also have two possibilities. The endpoints can be given
as two separate arguments, a and b, or can be combined into one vector argument,
[a,b]. The quadrature functions quad and quadl use two separate arguments. The
zero finder, fzero, uses a single argument because either a single starting point or
a two-element vector can specify the interval. The ordinary differential equation
solvers that we encounter in the next chapter also use a single argument because a
many-element vector can specify a set of points where the solution is to be evaluated.
The easy plotting function, ezplot, accepts either one or two arguments.
6.5 Performance
The Matlab demos directory includes a function named humps that is intended
to illustrate the behavior of graphics, quadrature, and zero-finding routines. The
function is
The statement
h(x) =
ezplot(@humps,0,1)
1
(x − 0.3) 2 + 0.01 +
1
(x − 0.9) 2 + 0.04 .
produces a graph of h(x) for 0 ≤ x ≤ 1. There is a fairly strong peak near x = 0.3
and a more modest peak near x = 0.9.
The default problem for quadgui is
quadgui(@humps,0,1,1.e-4)
You can see in Figure 6.3 that with this tolerance, the adaptive algorithm has
evaluated the integrand 93 times at points clustered near the two peaks.
With the Symbolic Toolbox, it is possible to analytically integrate h(x). The
statements
syms x
h = 1/((x-.3)^2+.01) + 1/((x-.9)^2+.04) - 6
I = int(h)
produce the indefinite integral
I = 10*atan(10*x-3)+5*atan(5*x-9/2)-6*x
The statements
D = simple(int(h,0,1))
Qexact = double(D)
produce a definite integral